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Chris Langan has sent in much material, most of which concerns his theory, CTMU, and its implied solutions of several math and logic problems. Some of his stuff has been criticized for being too demanding for the casual reader. As I was one of the critics, I was very pleased to receive, several months ago, a lengthy but entertaining article from Langan concerning a fictional scam-artist's analysis of the marble problem.
The article was long enough to comprise an entire issue all by itself, which also pleased me, since I run increasingly far behind in putting this journal together. Once I decided to make Langan's article into an issue of its own, more delays ensued. Chris Cole suggested that Langan send the disk on which he'd written the article, since that would save me from having to type the entire thing into the computer. We then wrote several letters back and forth concerning his disk's compatibility with Cole's system. We were not always prompt in our correspondence.
Here, finally, is Langan's story. I hope you enjoy it. If not, another issue should follow shortly.
by C.M. Langan
Those of you still with the group know me only as the author of the CTMU, a complex theory describing the nature of reality. Few of you have been sufficiently impressed with the theory to do more than criticize it in ways that suggest you have read nothing about it. While I have already used the CTMU for applications that would make the careers of any team of credentialed signatories -- granted, they would have to enlarge on certain aspects, but they would also be writing for a specialty journal in which such detail is welcome and appropriate -- I have been rewarded only by silence and unpopularity. This raises the question of why I continue to bother with your opinions.
The answer has several parts. First, it is against my nature to let people sell themselves short, and anyone who persistently disputes the given applications is ultimately going to look less intelligent than he now assumes he is. Next, I have a stake in the journal, having used it to introduce the theory. You all had a part in that, and -- despite your limited uptake -- must be treated accordingly. Next, I keep hearing about the unlimited potential of this group to make a mark in the annals of intellectual history, if not to solve problems like P?NP, the four-color map problem, and the several problems whose solutions I've already published. If I had no credence in this potential, I would never have used Noesis for anything but mild entertainment, which is all that the majority of members seem to get out of it. And last but not least, the CTMU is of great intrinsic importance. In a sense truer than many of the weird cosmologies propagated in professional journals, it promises to be an "ultimate theory of everything". If you still doubt it, I suggest that you finally put yourself to the trouble of reading the issues I edited...even if it means having to wake up and spill a little coffee on them.
Granted, I used a few too many big words. Mea culpa. But when trying to introduce any theory in a limited number of words, one must often resort to very dense language. Even so, I kept the dense passages to a minimum and used as many illustrations as I could fit between them. The end result is that for every objection made by any of you, I can cite several passages from Noesis 44-49 in light of which it should not have been made. This is a blanket observation which applies to every criticism offered thus far. To object is fine, but you have to be able to listen to reason.
Let's take an example. Those of you who read Noesis only casually are aware of the recent criticisms of Messrs. Cole and Dicks, both of whom object to "inadequacies" in the CTMU. Consider those involving probability theory. Chris says that probabilistic computations are impossible without very specific kinds of prior information. George claims that probabilistic computations are necessary and should therefore be undertaken regardless of how little information is available. A contradiction plainly exists between these viewpoints. But rather than criticizing each other, they both criticize the CTMU, which is -- for reasons long since given -- the only formalism in which it is possible to reconcile their views. Probabilistic paradoxes do indeed exist, and they are utterly inimical to any naive (non-metamathematical) approach to empirical induction. Yet, it's me who gets caught in the crossfire, despite the CTMU resolvability of Chris's paradoxes and the CTMU formalization of George's style of induction! This situation is even more pronounced with respect to Newcomb's paradox. If not for the CTMU, the fur would still be flying among the partisans of these two members.
Nonetheless, I have managed to use your objections in a way that maximizes their value. Any creative researcher who qualifies as more than a drone faces two kinds of problem. First, he must solve the problem his research is designed to address. And then he must figure out how to finesse it past the majority of his self-serving, mutually negative, and deliberately obtuse "peers", who can always be expected to prefer their former viewpoints no matter how weak, inadequate, or paradox-ridden they may have been. The first problem is logical, mathematical, and scientific. The second is psychological and political...in short, human. Even the most talented and intelligent thinkers, who are easily up to the first phase of this process, are dismayed and disoriented by the next. And I'm talking about people who are for the most part already accepted within their disciplines. If they weren't -- as I am not -- many of them would fold up and start selling insurance.
The premature objections thus far leveled at the CTMU and its applications in Noesis are valuable in that they indicate those parts of the theory which, having been initially unpalatable to high-IQ readers, are likely to stick in the craws of the public at large. They can thus be specifically targeted in the book I'm writing to explain the theory in detail. In the language of the CTMU itself, they will let me focus on particular deficiencies in the acceptive and inferential syntaxes of the â-sub-automata for whom the book is being written. To those of you who have at least contributed to this extent, I give thanks. To the rest of you, I really didn't know what to give until the emergence of Rick Rosner suggested a humorous approach to the "debate".
Rick, who admittedly walked in late on all this, claims that he doesn't "believe in" my solution to the infamous marble problem or in The Resolution of Newcomb's Paradox. Rick is too intelligent to make this denial, given my previous explanations. I'm therefore left to assume that he glossed over them, possibly because they don't conform to his idea of good reading. This has compelled me to undertake a total revision of my stilted, pedantic, and long-winded writing style. That is, where I'd been using learned allusions and 99 cent jawbreakers, I've taken to using colorful images, cheap jokes, and antediluvian cliches. Alright, so maybe it is beneath the dignity of my ideas. But hey, you've got to get your points across, and maybe this is the way to do it. So Goodbye, English language, and Hello, clown language.
I resolved Newcomb's Paradox. You quibbled. I settled the crisis in physics. You napped. I clarified the process of logical and empirical induction. You snored. Nothing I could say made the slightest difference in your fixed opinions concerning the issues in question. I have a name, am probably only human, and therefore could not possibly be smart enough to tell you anything. My voice simply doesn't carry that far over the clouds. But even though you tuned me clean out of your elevated awareness, there may yet be one whom you will not ignore. One who is so smart, so endearing, and so outrageously entertaining that even the stones grow ears at the sound of his name. I gave you six issues of Chris Langan, a character who failed abysmally to tickle your fancy. Well, you ain't seen nothin' yet, so get ready and get set. Because now, without further ado, from the steel and concrete warrens of the Big Apple, I bring you our new mascot, the unique, the amazing... Jojo Einstein, Street Clown. But be forewarned: what he's gonna do to the marble problem, he can do as easily to Newcomb's Paradox.
Now heeeeeere's Jojo!
A DAY IN THE LIFE OF JOJO EINSTEIN, STREET
By C.M. Langan
Any resemblances to real persons living or dead, et cetera. Jojo Einstein is fictional and his narrative need not reflect the views of the author, who nonetheless appreciates his apparent adulation.
It was autumn in Manhattan, that special season in the center of the universe. Jojo the clown, despite his sensitivity to life's subtler charms, was focussed in another direction: his next meal. Tourism was off now that the rubes had used up their summer vacations. Conventioneers and business people were abundant as always, but yuppies were not Jojo's stock in trade. He needed the kind of gullible backwoods yokel for whom the city was primarily a giant carnival, and summer was their time. Jojo, curly red wig vibrating in the gentle breeze, needed a mark, and needed one bad.
It was still warm enough for Fishman's Kosher Deli to have its doors open. Aromatic tendrils of fresh lox and knishes wafted into his nostrils, bunching as they traveled downward into clawed fists which mauled his tortured stomach mercilessly beneath the puffy, billowing cloth of his enormously polka-dotted jumpsuit. His belly was as empty as the toes of his custom-cobbled size 33-Z floppy shoes, now so worn that he could feel every bump in the grimy pavement through their perforated, paper-thin soles.
Jojo stared disgustedly at the sign he had taped to the brown brick facade of the deli. "MOMS!" it read through a colorful cloud of spray painted balloons, "How would your kids like to meet a REAL CIRCUS CLOWN? Available Now for Parties." This was followed by the telephone number of a combination flophouse-bordello in Hell's Kitchen, with "Please Leave Message" written beneath it. As he peeled loose the sign, he thought spitefully about adding a little something -- like (applications welcome) -- next to the number.
A Wall Street cowboy, Stetson cocked, blundered into Jojo as he inserted the rolled-up sign into his suit. "Hey hey, hey!", barked the enraged clown. "That's right, you. Ivan Milk-can! What's the matter," he yelled at the man's retreating back, "no time to talk? Okay then, step lively! Maybe you'll kick yourself in the tuchis." Jojo invisibly flexed his large muscles, reminding himself to get back into the gym as soon as he had worked out this little problem with his diet. He had doubled as a strongman in the circus, and still looked like a bodybuilder underneath his costume. The street was a jungle, and a clown had to be able to defend himself.
A mime slowed down as he approached, obviously sniffing around for a likely spot to do his routine. "Get lost!" snarled Jojo, not yet ready to relinquish his street corner. The mime scurried away, doing a furtive impression of the irascible clown. If there was one thing that bugged Jojo, it was a street mime poaching on his rightful territory. Pickings were slim enough, and these mimes were everywhere. Street musicians came in a close second at the bottom of his hit parade, but he wasn't above charging them "rent" for the privilege of providing him with a soundtrack. He was doing them a favor; even after his cut, they still made more than they could make without his visuals. It all came down to one simple truth: like the lion of the savanna and the shark of the sea, the clown was at the pinnacle of the street-showbiz food chain. And Jojo Einstein was the undisputed king of the street clowns.
Jojo's eyes narrowed into a hungry, grease-painted squint as they scanned the midday crowd on Times Square. Most of the faces were locked up tight, offering no chink through which a famished clown might access what remained of their naivete. No emotion, only motion. Then, over the traffic, he saw what he was looking for. It was nothing especially obvious, but Jojo's street-trained eyes had seen enough. What they saw was better than stupidity, better than gullibility. It was overconfidence.
With surprising quickness, the canny clown shuffled directly over the curb and jaywalked into the melee of scratched and dented cars. Brakes screeched as a taxi halted inches from him. The cabby swore viciously, but Jojo -- who would ordinarily have leapt onto the hood and done his patented hot dog dance for the amusement of onlookers -- took only enough time to smear a small shaving cream pie over the windshield before jouncing onward. He preferred to use Kreemi-Whip, but had eaten the last of that a while ago. He had sucked it directly out of the can, and it had blown so many of his gaskets that his colon had sustained a third-degree sunburn.
"Hey!" he shouted as he closed the gap on his quarry. "Hey mister, I think you dropped this!" "This" was a wallet, which Jojo held forth as the man looked back over his shoulder. "It's yours", Jojo assured him as he ambled closer. "See, it has your picture in it!" He snapped the wallet open in the young man's face, releasing a salvo of small paper flowers. The youth, trying to spit away a blossom which had stuck to his lips, surveyed Jojo with a mixture of surprise and annoyance.
"Now that I've got your attention, sir", said Jojo, talking fast as he waddled comically around to face his victim, "I couldn't help noticing that you look like a fellow who isn't afraid to take a little risk every now and again. A betting man, if you will!" He had positioned himself directly in the mark's path, preventing escape. The mark just stood there, his briefcase dangling at his side. A logo, "World's Most Exclusive IQ Society", was tastefully inscribed on the mauve vinyl. There was the bait, and man, was it ever perfect!
"You have a very intelligent face. Anyone ever tell you that?" The egghead blinked. "Quite intelligent! Why, I'd bet your IQ is right through the roof! Am I right?"
"Look here," said the whiz-kid. "I've got a meeting in..."
"I won't take much of your time. No siree, bob. I just want you to take a little IQ test, nothing really for a guy like you. You can even win a little money, one buck gets you a dollar, five gets you ten."
"Well, I really don't..."
"Aw, c'mon. You're not afraid to use your smarts to make a little dough, are you?" Jojo pulled a fist from his pocket, held it under the kid's nose, and opened it for a fraction of a second. The kid glimpsed an innocent-looking pink polyhedron. "This is a die, a fair die. But instead of the usual six faces, this one has twelve. A regular die's a cube. This one's something different, I forget the name." Jojo had to suppress a smile as he feigned a tip-of-the-tongue word search. "What do they call it, a, a..."
"A dodecahedron," the kid finished on cue, impaling himself firmly on the hook. "One of the five Platonic solids."
"Yeah!" said Jojo gratefully. "That's it, a dodecahedron. Man, I sure wish I had your IQ. This is gonna be so easy for you!" He pulled a narrow open-ended tube from one of his many pockets and deposited the die inside it. "See, the game is, you shake it once and look at the top number. Then you bet that on the next shake, a different number shows up. One buck gets you a dollar, five gets you ten." Jojo didn't have a red cent, but this looked to be a lead-pipe cinch.
The mark sighed resignedly, digging into his pants pocket. He extracted several bills, the smallest of which was a five. Jojo gingerly plucked out a fin. "You won't regret it, sir, I can promise you that!"
"Okay", said the genius. "It's a fair die. That means that whatever face shows up on the first shake has a one-in-twelve chance of showing on the second shake. I have an eleven-in-twelve chance of winning. Those are great odds, so let's go." He accepted the tube and gave it a perfunctory shake before looking inside. A tiny portrait of Jojo grinned up at him, eyes crossed towards the number "1" where a nose should have been. He showed it to the clown.
"Why, it's me!" said Jojo delightedly.
"So it is", said the kid, too confidently. "But it's one-in-twelve that you'll survive the next shake with the same olfactor." He shook again, this time more sincerely.
"Ready to win?" said Jojo, barely able to suppress a burst of untimely laughter.
"As a matter of fact, I am", said the mark. He stopped shaking and peered into the tube. His expression turned dark.
"Lemme see", said Jojo, tilting the tube in his direction as the mark held onto it. Sure enough, there he was again, still with a big red "1" in place of his red rubber proboscis. "Aw, gee, sir, tough luck. Looks like you got a bad break. Want to go again? Five gets you ten, ten gets you twenty."
"Something's fishy here", said the kid with a sour-grapes edge to his voice. "You won't mind if I have a closer look at your 'fair die', will you?" He put his briefcase down on the sidewalk and inverted the tube over his open hand. The pink polyhedron spilled out. On every face was a picture of Jojo, a "1" in place of his snout.
"You're a crook", accused the kid. "Give me back my five dollars, please."
"Oh, ho, ho!" chortled Jojo, royally amused. "And why am I a crook, pray tell?" He wadded up the fiver, stretched his huge ruffled collar open with a finger, and dropped it down the opening.
"You said this was a fair die. It isn't."
"Really?" said Jojo with mock incredulity. "How do you figure?"
The kid was indignant. "The die isn't fair because every face has the same thing on it."
"Well, so it does!" agreed Jojo. "But look at it, sir. Doesn't each face have an equal chance to end up on top?"
The kid assumed a pained expression. "So what? Each face is supposed to be different!"
"Is that right! And who says that?" Jojo's mirth was rapidly approaching an uncontrollable level.
The kid looked Jojo up and down, contempt smeared unprettily across his features. Then he nodded with an I've-got-your-number air. "I see. Here, take your junk back. I'm busy."
"Why, certainly, sir." Jojo took the tube and die, remaining squarely in the kid's path. "But are you sure I can't interest you in another kind of game, one a little less tricky?"
The mark, too angry for his own good, could not believe that the clown thought he could be taken twice in a row. But that appeared to be the program. With ice in his voice, he bit again. "What other game?"
Jojo had already pulled a covered black box out of his suit. He shook it; it rattled. "There are ten marbles in this box. You pull out ten marbles one at a time, each time replacing the marble in the box. Then you tell me what colors are inside. Then we look. If you're right, you win. Five gets you ten, ten gets you twenty." The brain looked at the box, calculating. Finally he spoke. "Alright. But I'm betting twenty dollars. Agreed?" He was all fired up, red hot for revenge.
The clown had to struggle to free his spasming larynx from a violent urge to cackle. "Whatever you like!" He held out his hand. The kid fished out a Jackson and slapped it into his palm. Jojo raised the box to slightly above eye level and removed the lid. The mark reached up and into the box. When his hand emerged, a snow white marble glinted brilliantly between two fingers.
"White!" said Jojo enthusiastically. "That's good luck". The kid dropped the marble back into the box. Jojo lifted the box straight over his head and did an imitation of Chubby Checker imitating a belly dancer, thus mixing its contents. "That was trial one", he informed the pigeon. "You got nine more to go."
The second trial was identical. "Jeez!" commented the clown. "You're defyin' the laws of chance!"
Eight more times, the act was repeated. Eight more times, a white marble was pulled from the box.
"Well done, professor", congratulated Jojo. "Now what do you say? What colors of marbles are in the box?"
The mark smiled disdainfully, clearly bent on showing off. "I'll tell you. But first, let me tell you how I calculated the answer."
"Be my guest", said Jojo, his star blue eyes twinkling merrily.
"There are ten marbles in the box. That means ten possible colors, one for each marble. But that's only an upper limit, just like the number of faces on your dodecahedral die was an upper limit on the number of distinct symbols on the faces. There isn't any specific dependency of one on the other."
"What?" said the clown, feigning puzzlement.
"When you set an upper and lower limit on a random variable, you're only defining the range of the variable. If you tell me the number of colors is between one and ten inclusive, you're not giving me one iota of information about where the number falls within that range. The variable is still random."
"No kidding", wondered the clown.
"If I were to use that range in my calculation of the specific color distribution of marbles in the box, I'd be trying to get a kind of information out of it that it doesn't have. I'd be trying to convert general information directly into specific information. Any conclusion I might draw in the process would thus be invalid -- the same way my former conclusion about the die was invalid."
"Man oh man!" marveled Jojo. "What an IQ!"
Smugly, the genius continued. "It follows that my only source of information about the specific number of colors, and the exact distribution of marbles among those colors, is observation. That's pure empirical science!"
Jojo rolled his eyes and whistled softly in admiration.
"The limit is important, but only when you try to violate it. For instance, if I try to calculate the probability that ten solidly-colored marbles have eleven different colors, I get zero. If I'm using Bayes' rule, the limit is automatically reflected in the zero probability of any hypothesis surpassing it. That's how the limit figures in Bayesian inference...not as a quantifier of possibilities."
Jojo stared raptly, pretending to be hypnotized. If the kid had kept on babbling about ten possible colors after picking ten whities in a row, then he, Jojo, ordinarily as compassionate as a piece of earthmoving equipment, might have had to break down and steer him to a shrink. Before he could hurt himself.
"I observed just one color, white, in ten trials. Say I use Bayes' rule to calculate the probability of an all-white color distribution. If I start out with the idea that the only possible color is white, I naturally get one hundred percent."
"Cowabunga!" exclaimed Jojo. "That's not a probability. That's a certainty!" "Precisely", said the high-IQ wizard knowingly. "And certainty is exactly what sampling with replacement can't give you. So it's wrong to confine the range of possible colors to white alone. But Bayes' rule is still the right rule to use in a calculation of this kind. So I keep the rule, but throw out the idea of using only one possible color."
Jojo nodded, hanging worshipfully on the mark's every word.
"Any color besides white is clearly nonwhite. Yet, I have no evidence -- no observation -- that would justify a subdivision of 'nonwhite' into specific nonwhite colors like red and green. So by simple elimination, 'nonwhite' must be the ticket. I now have two possible colors, white and nonwhite. And the beauty of it is, the term 'nonwhite' consistently represents every possible color other than white! Nothing is excluded. And nothing is represented at a level of specificity unwarranted by observation."
"Wow!", sighed Jojo. The mark thought the clown was interested in what he had to say. In fact, Jojo could hear nothing over the silent strains of "Under the Big Top" rising triumphantly in his elephantine nerf ears. The ears were a little precaution he sometimes took when he thought it might turn cold; they doubled nicely as earmuffs.
"Bayes' rule is designed to update probabilities. It combines initial information with subsequent observation, and needs both to work. I just made ten observations. But I still need a set of initial probabilities to combine them with. Since I'm looking for a distribution of ten marbles among two colors, I need an initial probability for each of the ten possible distributions I might find. I knew nothing before I started sampling; I had no information on the distribution. But it takes information to distinguish anything from anything else, probabilities included. Without information, there's no basis for calling any possible distribution 'more probable' than any other. So I have to set all the initial probabilities equal. From this and observation, it follows that the probability of an all-white distribution is approximately .67. That makes it the most probable distribution. So I estimate that all ten marbles in the box are white!" He waited expectantly.
Jojo's mouth sadly parodied the gaping grin painted around it. He lowered the box. The mark looked inside. As Jojo tilted the box gently in the mark's direction, a row of immaculate white marbles collected along the bottom edge.
The kid looked up, confused. "I count only nine marbles. Where's the tenth?"
Jojo tapped the side of the box. A solitary black marble rolled down to join the white ones. The mark dropped his jaw, totally deflated.
"You know, maybe it just isn't your day", consoled the clown, replacing the box in his garment and pocketing the twenty. "I'm not even gonna ask you if you want to try again." He could already taste the bagels, prosciutto, and ice cream he was going to be eating as soon as he could lose the sucker. "But man, are you ever brilliant! I'd ask you for your autograph, but I know you're busy." Excess salivation was making it hard for him to enunciate. Otherwise, he might have explained that the city had a couple too many bridges, and had appointed him exclusive cut-rate broker for one down Brooklyn way.
"C'est la vie", sighed the genius, bending down for his briefcase. "Wait. Better make that solvitur ambulando." Crestfallen but smiling ruefully, he plodded tiredly away to his just oblivion.
As Jojo stuffed his face in the sit-down deli half a block away, he reflected on the score. First, the mark had been a real dope to assume that there was some mathematical law relating the number of faces of a "fair die" to what's on them. The contents of faces are totally independent of the number of faces except at the very limit -- that is, on the verge of logical inconsistency -- and "total independence from" means "no information on". When two variables are unrelated by definition, any such relation holding in a given case can only be confirmed by observation. Dependency relations are subject to the same criterion as everything else in probability theory: no confirmee, no usee.
Then the kid had wised up with the marbles, which was almost the same problem with a ten-sided "die" and colors instead of numbers. But he hadn't wised up enough. Almost all clowns know some magic, and Jojo had been a close-up magician for years. The sleight of hand by which he'd inserted the black marble had been no big deal. Just goes to show you, he ruminated: there's a sucker born every minute, a fool is soon separated from his cash, and what's more, nobody ever lost any money underestimating the intelligence of the American public. Man, what he could have done to that kid with a deck of cards!